This work is concerned with the dynamics of online cultural markets, namely, attention allocation of many users on a set of digital goods with infinite supply. Such dynamic is important in shaping processes and outcomes in society, from trending items in entertainment, collective knowledge creation, to election outcomes. The outcomes of online cultural markets are susceptible to intricate social influence dynamics, particularly so when the community comprises consumers with heterogeneous interests. This has made formal analysis of these markets improbable. In this paper, we remedy this by establishing robust connections between influence dynamics and optimization processes, in trial-offer markets where the consumer preferences are modelled by multinomial logit. Among other results, we show that the proportional-response-esque influence dynamic is equivalent to stochastic mirror descent on a convex objective function, thus leading to a stable and predictable outcome. When all consumers are homogeneous, the objective function has a natural interpretation as a weighted sum of efficiency and diversity of the culture market. In simulations driven by real-world preferences collected from a large-scale recommender system, we observe that ranking strategies aligned with the underlying heterogeneous preferences are more stable, and achieves higher efficiency and diversity. In simulations driven by real-world preferences collected from a large-scale recommender system, we observe that ranking strategies aligned with the underlying heterogeneous preferences are more stable, and achieves higher efficiency and diversity.